Abstract
Analyzing multidimensional data is a fundamental problem in various areas of computer science. As the amount of data is often huge, it is desirable to obtain sublinear time algorithms to understand local properties of the data.
We focus on the natural problem of testing pattern freeness: given a large ddimensional array A and a fixed ddimensional pattern P over a finite alphabet Gamma, we say that A is Pfree if it does not contain a copy of the forbidden pattern P as a consecutive subarray. The distance of A to Pfreeness is the fraction of the entries of A that need to be modified to make it Pfree.
For any epsilon > 0 and any large enough pattern P over any alphabet  other than a very small set of exceptional patterns  we design a tolerant tester that distinguishes between the case that the distance is at least epsilon and the case that the distance is at most a_d epsilon, with query complexity and running time c_d epsilon^{1}, where a_d < 1 and c_d depend only on the dimension d. These testers only need to access uniformly random blocks of samples from the input A.
To analyze the testers we establish several combinatorial results, including the following ddimensional modification lemma, which might be of independent interest: For any large enough ddimensional pattern P over any alphabet (excluding a small set of exceptional patterns for the binary case), and any ddimensional array A containing a copy of P, one can delete this copy by modifying one of its locations without creating new Pcopies in A.
Our results address an open question of Fischer and Newman, who asked whether there exist efficient testers for properties related to tight substructures in multidimensional structured data.
BibTeX  Entry
@InProceedings{beneliezer_et_al:LIPIcs:2017:7442,
author = {Omri BenEliezer and Simon Korman and Daniel Reichman},
title = {{Deleting and Testing Forbidden Patterns in MultiDimensional Arrays}},
booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
pages = {9:19:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770415},
ISSN = {18688969},
year = {2017},
volume = {80},
editor = {Ioannis Chatzigiannakis and Piotr Indyk and Fabian Kuhn and Anca Muscholl},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7442},
URN = {urn:nbn:de:0030drops74427},
doi = {10.4230/LIPIcs.ICALP.2017.9},
annote = {Keywords: Property testing, Sublinear algorithms, Pattern matching}
}
Keywords: 

Property testing, Sublinear algorithms, Pattern matching 
Collection: 

44th International Colloquium on Automata, Languages, and Programming (ICALP 2017) 
Issue Date: 

2017 
Date of publication: 

07.07.2017 